Assumptions

  1. Linearity
  2. Independence of Errors
  3. Normality of Errors
  4. Equal Variance of Errors

Formulas

\[ Y_i = \beta_0 + \beta_1X_i + \epsilon_i \]

\[ \hat{Y}_i = \beta_0 + \beta_1X_i \] * Error or Residual

\[ \epsilon_i = Y_i - \hat{Y}_i \]

\[ b_1 = \frac{SSXY}{SSX} \]

where

\[ SSXY = \sum_{i=1}^{n}(X_i-\bar{X})(Y_i-\bar{Y}) \]

=

\[ \frac{\sum_{i=1}^{n}(X_i)\sum_{i=1}^{n}(Y_i)}{n} \] * Intercept

\[ b_0 = \bar{Y}+b_1\bar{X} \]

\[ SST = \sum_{i=1}^{n}(Y_i-\bar{Y})^2 \]

\[ SSR = \sum_{i=1}^{n}(\hat{Y}_i-\bar{Y})^2 \]

\[ SSE = \sum_{i=1}^{n}(Y_i-\hat{Y}_i)^2 \]

\(r^2\) is always between 0 and 1 \(0 \le r^2 \le 1\)

\[ r^2 = \frac{SSR}{SST} = \]

\[ 1 - \frac{SSE}{SST} \]

\[ S_{YX} = \sqrt{\frac{SSE}{n-2}} \]

\[ S_{b_1} = \frac{S_{YX}}{\sqrt{SSX}} \]

\[ \hat{Y} \pm t_{\frac{\alpha}{2}}S_{YX}\sqrt{h_i} \]

\[ \hat{Y} \pm t_{\frac{\alpha}{2}}S_{YX}\sqrt{1+ h_i} \] where

\[ h_i = \frac{1}{n}+\frac{(X_i - \bar{X})^2}{SSX} \] \[ h_i = \frac{1}{n}+\frac{(X_i - \bar{X})^2}{\sum(X_i - \bar{X})^2} \]

\[ F_{STAT} = \frac{MSR}{MSE} \] \[ MSR = \frac{SSR}{k} \] \[ MSE = \frac{SSE}{n-k-1} \]

t-Test for the Slope

\(H_0\): \(\beta_1\) = 0

\(H_1\): \(\beta_1\) \(\ne\) 0

\(t_{STAT}\) = \(\frac{b_1-\beta_1}{S_{b_1}}\)

\(d.f.\) = n-2

Confidence Interval Estimate for the Slope

\[ b_1 \pm t_{\frac{\alpha}{2}}S_{b_1} \] and

\[ d.f. = n-2 \]

t-Test for Correlation Coefficient

\(H_0\): \(\rho\) = 0

\(H_1\): \(\rho\) \(\ne\) 0

\(t_{STAT}\) = \(\frac{r - \rho}{\sqrt{\frac{1-r^2}{n-2}}}\)

\(d.f.\) = n-2

\(r\) = + \(\sqrt{r^2}\) if \(b_1 > 0\) or \(r\) = - \(\sqrt{r^2}\) if \(b_1 < 0\)

#install.packages("UsingR")
library(UsingR)
## Warning: package 'UsingR' was built under R version 4.1.2
## Loading required package: MASS
## Warning: package 'MASS' was built under R version 4.1.2
## Loading required package: HistData
## Loading required package: Hmisc
## Warning: package 'Hmisc' was built under R version 4.1.2
## Loading required package: lattice
## Loading required package: survival
## Warning: package 'survival' was built under R version 4.1.2
## Loading required package: Formula
## Loading required package: ggplot2
## Warning: package 'ggplot2' was built under R version 4.1.2
## 
## Attaching package: 'Hmisc'
## The following objects are masked from 'package:base':
## 
##     format.pval, units
## 
## Attaching package: 'UsingR'
## The following object is masked from 'package:survival':
## 
##     cancer
## Load Data Set
data(galton)

## Create a Scatter Plot

plot(galton$parent,galton$child,pch=19,col="blue")

## Conduct a linear Model Regression
  
lm1 <- lm(galton$child~galton$parent)

## Add Regression Line to the scatter plot

lines(galton$parent, lm1$fitted, col="black",lwd=3)

## Create Sample Record

newGalton <- data.frame(parent=rep(NA,1e6), child=rep(NA,1e6))

## Verify Record

head(newGalton)
##   parent child
## 1     NA    NA
## 2     NA    NA
## 3     NA    NA
## 4     NA    NA
## 5     NA    NA
## 6     NA    NA
## Add Parent Records
newGalton$parent <- rnorm(1e6,mean=mean(galton$parent), sd=sd(galton$parent))

## Add Child Record

newGalton$child <- lm1$coeff[1] + lm1$coeff[2]*newGalton$parent + rnorm(1e6,sd=sd(lm1$residuals))

## Verify New Data Set

head(newGalton)
##     parent    child
## 1 66.33289 65.50516
## 2 65.85542 66.14508
## 3 66.16100 71.81595
## 4 68.22543 66.81991
## 5 67.67480 66.96356
## 6 68.91756 69.74420
## Visualize
smoothScatter(newGalton$parent,newGalton$child)
abline(lm1,col="red",lwd=3)

## Start Taking Samples:
set.seed(12345)
sampleGalton1 <- newGalton[sample(1:1e6,50,replace=F),]
plot(sampleGalton1,pch=19,col="blue")
sampleLm1 <- lm(sampleGalton1$child~sampleGalton1$parent)
lines(sampleGalton1$parent,sampleLm1$fitted,lwd=3,lty=2)
abline(lm1,col="red",lwd=3)

## Take another Sample
sampleGalton2 <- newGalton[sample(1:1e6,50,replace=F),]
plot(sampleGalton2,pch=19,col="blue")
sampleLm2 <- lm(sampleGalton2$child~sampleGalton2$parent)
lines(sampleGalton1$parent,sampleLm1$fitted,lwd=3,lty=2)
lines(sampleGalton2$parent,sampleLm2$fitted,lwd=3,lty=3)
abline(lm1,col="red",lwd=3)

## Take many samples

sampleLm <- vector(100,mode="list")
for(i in 1:100){ sampleGalton <- newGalton[sample(1:1e6,50,replace=F),];sampleLm[[i]]<- lm(sampleGalton$child~sampleGalton$parent)}

##n Visualize

smoothScatter(newGalton$parent,newGalton$child)
for(i in 1:100){abline(sampleLm[[i]],lwd=3,lty=2)}
abline(lm1,col="red",lwd=4)

## Check Histogram of the Coefficients

par(mfrow=c(1,2))
hist(sapply(sampleLm,function(x){coef(x)[1]}),col="blue",xlab="Intercept",main="")
hist(sapply(sampleLm,function(x){coef(x)[2]}),col="blue",xlab="Slope",main="")

summary(sampleLm2)$coeff
##                        Estimate Std. Error  t value     Pr(>|t|)
## (Intercept)          29.9371244  8.2120274 3.645522 6.550151e-04
## sampleGalton2$parent  0.5589164  0.1203513 4.644041 2.678252e-05
confint(sampleLm2,level=0.95)
##                           2.5 %     97.5 %
## (Intercept)          13.4257366 46.4485121
## sampleGalton2$parent  0.3169339  0.8008989
## End of First Regression!!!! 


# Get and load library FPP
library("fpp")
## Loading required package: forecast
## Warning: package 'forecast' was built under R version 4.1.2
## Registered S3 method overwritten by 'quantmod':
##   method            from
##   as.zoo.data.frame zoo
## Loading required package: fma
## 
## Attaching package: 'fma'
## The following object is masked from 'package:UsingR':
## 
##     chicken
## The following objects are masked from 'package:MASS':
## 
##     cement, housing, petrol
## Loading required package: expsmooth
## Loading required package: lmtest
## Warning: package 'lmtest' was built under R version 4.1.2
## Loading required package: zoo
## Warning: package 'zoo' was built under R version 4.1.2
## 
## Attaching package: 'zoo'
## The following objects are masked from 'package:base':
## 
##     as.Date, as.Date.numeric
## Loading required package: tseries
## Warning: package 'tseries' was built under R version 4.1.2
# Loop at all the variables 

plot(fuel[,-1])
pairs(fuel[,-1])

summary(fuel)
##                              Model       Cylinders         Litres    
##  Volvo S60 FWD                  :  3   Min.   :4.000   Min.   :1.30  
##  Chevrolet Colorado 2WD         :  2   1st Qu.:4.000   1st Qu.:2.00  
##  Chevrolet Colorado Crew Cab 2WD:  2   Median :4.000   Median :2.40  
##  Chevrolet Malibu               :  2   Mean   :4.157   Mean   :2.31  
##  Chrysler PT Cruiser            :  2   3rd Qu.:4.000   3rd Qu.:2.50  
##  Dodge Caliber                  :  2   Max.   :5.000   Max.   :3.70  
##  (Other)                        :121                                 
##     Barrels           City          Highway           Cost     
##  Min.   : 7.40   Min.   :15.00   Min.   :20.00   Min.   : 615  
##  1st Qu.:13.20   1st Qu.:19.00   1st Qu.:27.00   1st Qu.:1091  
##  Median :14.30   Median :21.00   Median :28.00   Median :1182  
##  Mean   :14.31   Mean   :21.97   Mean   :28.89   Mean   :1185  
##  3rd Qu.:15.60   3rd Qu.:23.75   3rd Qu.:31.00   3rd Qu.:1290  
##  Max.   :20.10   Max.   :48.00   Max.   :45.00   Max.   :1667  
##                                                                
##      Carbon      
##  Min.   : 4.000  
##  1st Qu.: 7.100  
##  Median : 7.700  
##  Mean   : 7.671  
##  3rd Qu.: 8.300  
##  Max.   :10.800  
## 
cor(fuel[,-1])
##            Cylinders     Litres    Barrels       City    Highway       Cost
## Cylinders  1.0000000  0.5072599  0.3612070 -0.3246731 -0.2863089  0.3615217
## Litres     0.5072599  1.0000000  0.6795817 -0.5228023 -0.6663893  0.6796274
## Barrels    0.3612070  0.6795817  1.0000000 -0.9062413 -0.9288892  0.9999567
## City      -0.3246731 -0.5228023 -0.9062413  1.0000000  0.8209571 -0.9052234
## Highway   -0.2863089 -0.6663893 -0.9288892  0.8209571  1.0000000 -0.9284566
## Cost       0.3615217  0.6796274  0.9999567 -0.9052234 -0.9284566  1.0000000
## Carbon     0.3633336  0.6798169  0.9994928 -0.9079411 -0.9267689  0.9994937
##               Carbon
## Cylinders  0.3633336
## Litres     0.6798169
## Barrels    0.9994928
## City      -0.9079411
## Highway   -0.9267689
## Cost       0.9994937
## Carbon     1.0000000
# Assumption of Normality
hist(fuel$Carbon)

# Assumption of Linearity

plot(Carbon ~ City, data = fuel)

# Lets do a plot of two variables

plot(jitter(Carbon) ~ jitter(City), xlab="City (mpg)", ylab="Carbon footprint (tonnes per year)", data=fuel)

# Simple Linear Regression

fit <- lm(Carbon ~ City, data=fuel)
fit
## 
## Call:
## lm(formula = Carbon ~ City, data = fuel)
## 
## Coefficients:
## (Intercept)         City  
##      12.526       -0.221
summary(fit)
## 
## Call:
## lm(formula = Carbon ~ City, data = fuel)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -0.7014 -0.3643 -0.1062  0.1938  2.0809 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 12.525647   0.199232   62.87   <2e-16 ***
## City        -0.220970   0.008878  -24.89   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4703 on 132 degrees of freedom
## Multiple R-squared:  0.8244, Adjusted R-squared:  0.823 
## F-statistic: 619.5 on 1 and 132 DF,  p-value: < 2.2e-16
anova(fit)
## Analysis of Variance Table
## 
## Response: Carbon
##            Df  Sum Sq Mean Sq F value    Pr(>F)    
## City        1 137.005 137.005  619.52 < 2.2e-16 ***
## Residuals 132  29.191   0.221                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#Draw the regression line in the plot abline(fit)
# Lets do a residual plot and see what it looks like

res <- residuals(fit)
plot(jitter(res)~jitter(City), ylab="Residuals", xlab="City", data=fuel)

abline(0,0)

# Lets do some plots to see what R gives for regression plot(fit)
# On with Forecasting to see the output from regression

fitted(fit)[1]
##       20 
## 7.001388
# ask to do a forecast for predictor = 30

fcast <- forecast(fit, newdata=data.frame(City=30))

# plot the forecast to see the graph

plot(fcast, xlab="City (mpg)", ylab="Carbon footprint (tons per year)")

# Lets ask for some confidence interval for our forecast

confint(fit,level=0.95)
##                  2.5 %     97.5 %
## (Intercept) 12.1315464 12.9197478
## City        -0.2385315 -0.2034092
# Lets do a transformation of response and predictors to get a better fit. 
# Split the plotting output area into 2 columns and 1 row. 

par(mfrow=c(1,2))

# Do a Regression on log of the data.

fit2 <- lm(log(Carbon) ~ log(City), data=fuel)
# Plot 1

plot(jitter(Carbon) ~ jitter(City), xlab="City (mpg)",
ylab="Carbon footprint (tonnes per year)", data=fuel)

# Add a line based on the exponents of the fit variables

lines(1:50, exp(fit2$coef[1]+fit2$coef[2]*log(1:50)))

# Plot 2

plot(log(jitter(Carbon)) ~ log(jitter(City)),
xlab="log City mpg", ylab="log carbon footprint", data=fuel) 
 
#Draw the regression line

abline(fit2)

# Look at the residuals.
#Store the residuals in a variable

res <- residuals(fit2)

# Lets so the residuals a little bigger 
par(mfrow=c(1,1))
plot(jitter(res, amount=.005) ~ jitter(log(City)),
ylab="Residuals", xlab="log(City)", data=fuel)

# End of Simple Linear Regression Exercise

# Multiple Linear Regression Example View(credit)

# Get Correlation Matrix
cor(credit)
##                     score     savings      income         fte      single
## score          1.00000000  0.40440532  0.31005351  0.05934809 -0.09301482
## savings        0.40440532  1.00000000  0.13754620 -0.03330092 -0.08252345
## income         0.31005351  0.13754620  1.00000000  0.12486839 -0.35237334
## fte            0.05934809 -0.03330092  0.12486839  1.00000000  0.07021355
## single        -0.09301482 -0.08252345 -0.35237334  0.07021355  1.00000000
## time.address   0.15033646  0.02273133 -0.01311510  0.05959628  0.03527295
## time.employed  0.17972354 -0.04713921  0.06951179 -0.04203080 -0.07679786
##               time.address time.employed
## score           0.15033646    0.17972354
## savings         0.02273133   -0.04713921
## income         -0.01311510    0.06951179
## fte             0.05959628   -0.04203080
## single          0.03527295   -0.07679786
## time.address    1.00000000    0.05336579
## time.employed   0.05336579    1.00000000
# Look at the correlation visually 

pairs(credit[,-(4:5)])

# Plot reveals that we need to do a transformation. As some values are zero we will do log +1
#Lets create a new variable with the logs. 

creditlog <- data.frame(score=credit$score, log.savings=log(credit$savings+1), log.income=log(credit$income+1), log.address=log(credit$time.address+1), log.employed=log(credit$time.employed+1), fte=credit$fte, single=credit$single)

# Now look at correlation 

pairs(creditlog[,1:5])

# Lets do Multiple Regression and do a fit

mfit <- step(lm(score ~ log.savings + log.income + log.address + log.employed + single, data=creditlog))
## Start:  AIC=2325.38
## score ~ log.savings + log.income + log.address + log.employed + 
##     single
## 
##                Df Sum of Sq   RSS    AIC
## - single        1      40.6 51132 2323.8
## <none>                      51091 2325.4
## - log.employed  1    1052.9 52144 2333.6
## - log.income    1    1306.9 52398 2336.0
## - log.address   1    3914.8 55006 2360.3
## - log.savings   1   29555.3 80647 2551.6
## 
## Step:  AIC=2323.78
## score ~ log.savings + log.income + log.address + log.employed
## 
##                Df Sum of Sq   RSS    AIC
## <none>                      51132 2323.8
## - log.employed  1    1063.7 52196 2332.1
## - log.income    1    1666.2 52798 2337.8
## - log.address   1    3891.0 55023 2358.4
## - log.savings   1   29515.1 80647 2549.6
summary(mfit)
## 
## Call:
## lm(formula = score ~ log.savings + log.income + log.address + 
##     log.employed, data = creditlog)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -26.133  -6.966  -1.125   5.379  37.446 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   -0.2186     5.2309  -0.042  0.96668    
## log.savings   10.3526     0.6124  16.904  < 2e-16 ***
## log.income     5.0521     1.2579   4.016 6.83e-05 ***
## log.address    2.6666     0.4345   6.137 1.72e-09 ***
## log.employed   1.3138     0.4094   3.209  0.00142 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 10.16 on 495 degrees of freedom
## Multiple R-squared:  0.4701, Adjusted R-squared:  0.4658 
## F-statistic: 109.8 on 4 and 495 DF,  p-value: < 2.2e-16
# Lets plot the original with the predicted. 
plot(fitted(mfit), creditlog$score, ylab="Score", xlab="Predicted score")

# Lets analyse Residuals.

# Lets do 2 rows two columns for our plot

par(mfrow=c(2,2))

plot(creditlog$log.savings,residuals(mfit),xlab="log(savings)") 

plot(creditlog$log.income,residuals(mfit),xlab="log(income)") 

plot(creditlog$log.address,residuals(mfit),xlab="log(address)") 

plot(creditlog$log.employed,residuals(mfit),xlab="log(employed)")

# One more plot of fitted and residuals
# Back to one column one row for big plots 

par(mfrow=c(1,1))

plot(fitted(mfit), residuals(mfit), xlab="Predicted scores", ylab="Residuals")

# Plot gives multiple graphs for regression analysis 
plot(mfit)

# More Outlier measures 
cooks.distance(mfit) 
##            1            2            3            4            5            6 
## 4.051051e-04 1.729785e-03 8.311866e-05 3.142736e-03 7.907947e-04 3.908466e-04 
##            7            8            9           10           11           12 
## 8.083366e-04 7.103351e-04 7.205655e-04 1.868111e-04 1.165014e-04 4.075184e-04 
##           13           14           15           16           17           18 
## 2.307393e-04 4.178442e-03 1.731300e-05 4.910764e-04 1.477818e-03 4.064875e-04 
##           19           20           21           22           23           24 
## 9.164389e-04 1.095508e-03 1.192237e-03 9.956692e-05 3.541666e-03 1.268629e-04 
##           25           26           27           28           29           30 
## 8.527395e-04 4.725106e-07 8.527052e-03 6.393144e-04 3.341921e-03 1.259718e-04 
##           31           32           33           34           35           36 
## 2.712541e-04 2.744761e-04 1.417882e-04 1.969763e-03 4.050582e-05 2.715615e-04 
##           37           38           39           40           41           42 
## 6.195392e-02 3.418682e-03 2.200256e-03 3.495179e-03 1.008214e-05 9.067949e-06 
##           43           44           45           46           47           48 
## 8.818308e-04 3.381141e-04 1.774477e-03 1.165606e-04 1.025643e-03 8.186106e-04 
##           49           50           51           52           53           54 
## 4.321428e-03 2.600804e-05 1.572226e-03 2.477833e-03 1.303792e-05 5.590926e-06 
##           55           56           57           58           59           60 
## 3.376027e-05 3.860727e-04 2.189096e-03 1.047668e-03 2.212494e-03 1.954115e-03 
##           61           62           63           64           65           66 
## 1.929580e-04 3.627637e-03 1.741508e-04 5.325141e-04 2.833372e-06 9.183896e-04 
##           67           68           69           70           71           72 
## 3.849327e-03 2.171143e-03 7.715362e-03 7.354513e-05 3.424648e-04 4.726752e-05 
##           73           74           75           76           77           78 
## 1.558689e-04 5.572702e-05 5.080879e-04 7.023998e-05 2.440131e-03 2.718839e-04 
##           79           80           81           82           83           84 
## 3.899619e-03 1.167024e-03 2.851297e-03 2.782971e-03 3.483400e-02 2.542610e-03 
##           85           86           87           88           89           90 
## 4.298478e-05 7.847650e-04 1.727130e-03 9.583978e-04 6.037729e-03 9.596773e-05 
##           91           92           93           94           95           96 
## 5.071991e-05 4.416070e-03 7.765058e-03 8.085673e-04 8.279679e-04 1.900803e-05 
##           97           98           99          100          101          102 
## 1.383593e-04 3.962356e-04 1.035818e-04 4.334311e-03 2.032776e-03 1.955998e-05 
##          103          104          105          106          107          108 
## 4.289720e-04 1.348355e-04 2.191473e-02 7.168164e-05 3.838467e-06 1.423444e-03 
##          109          110          111          112          113          114 
## 3.754928e-04 1.040292e-03 9.054161e-03 3.909952e-03 3.533724e-04 2.415057e-04 
##          115          116          117          118          119          120 
## 5.092441e-03 8.745124e-04 3.140147e-04 2.951805e-04 1.640536e-06 3.345009e-04 
##          121          122          123          124          125          126 
## 6.635401e-06 2.747851e-04 1.319360e-04 8.570672e-04 6.809195e-05 1.101615e-04 
##          127          128          129          130          131          132 
## 7.708570e-06 7.732614e-04 8.827675e-03 1.828411e-03 5.365437e-04 2.919913e-04 
##          133          134          135          136          137          138 
## 1.030097e-04 1.555994e-03 5.400540e-03 2.785334e-03 1.165650e-02 1.135483e-03 
##          139          140          141          142          143          144 
## 2.328636e-04 7.599702e-04 7.855314e-06 3.999202e-04 2.344337e-03 1.398212e-02 
##          145          146          147          148          149          150 
## 1.705616e-04 8.797003e-04 1.020859e-03 6.509040e-05 1.658737e-03 1.493473e-04 
##          151          152          153          154          155          156 
## 6.843117e-04 1.182026e-03 2.135121e-05 1.453412e-03 1.152734e-04 7.123579e-03 
##          157          158          159          160          161          162 
## 7.260600e-03 2.048427e-07 1.754778e-04 4.525302e-04 5.379148e-04 1.657840e-03 
##          163          164          165          166          167          168 
## 1.632431e-05 2.692196e-03 8.640181e-04 4.387531e-04 8.485658e-09 6.824317e-04 
##          169          170          171          172          173          174 
## 8.465303e-05 4.701519e-03 1.920932e-03 1.558927e-04 3.716776e-03 1.842968e-03 
##          175          176          177          178          179          180 
## 3.475816e-02 1.593831e-03 1.845392e-03 6.032396e-05 7.063502e-04 3.775858e-03 
##          181          182          183          184          185          186 
## 2.771270e-03 1.454387e-05 2.703894e-04 4.728581e-06 1.028516e-03 4.577951e-02 
##          187          188          189          190          191          192 
## 3.376428e-03 4.168602e-03 3.439168e-04 2.725357e-05 2.408123e-05 1.527414e-04 
##          193          194          195          196          197          198 
## 1.709611e-03 9.195786e-06 5.527057e-04 2.585964e-04 7.561515e-04 2.254806e-03 
##          199          200          201          202          203          204 
## 4.097967e-03 1.434187e-04 5.987985e-05 2.167177e-04 7.819290e-03 1.807388e-04 
##          205          206          207          208          209          210 
## 1.954493e-03 4.925005e-03 1.450325e-04 2.186163e-02 7.310055e-03 2.261924e-02 
##          211          212          213          214          215          216 
## 1.197311e-03 1.191412e-03 1.554084e-03 1.260414e-04 9.435582e-04 4.802544e-05 
##          217          218          219          220          221          222 
## 9.173002e-06 1.359986e-05 5.652554e-06 1.453128e-04 7.728299e-03 3.489421e-04 
##          223          224          225          226          227          228 
## 2.664422e-04 1.936458e-03 1.200312e-03 1.583171e-02 1.334945e-03 5.801202e-03 
##          229          230          231          232          233          234 
## 2.769853e-04 6.322407e-03 5.206445e-02 1.318283e-03 5.509224e-04 5.210090e-06 
##          235          236          237          238          239          240 
## 8.670782e-05 1.631937e-03 2.554565e-05 3.143729e-03 5.320888e-03 1.149965e-03 
##          241          242          243          244          245          246 
## 2.624935e-03 1.505782e-03 3.435900e-04 5.926477e-04 4.792521e-06 1.998259e-03 
##          247          248          249          250          251          252 
## 1.089049e-03 1.837863e-04 2.643252e-05 5.062546e-03 6.153077e-04 2.647678e-03 
##          253          254          255          256          257          258 
## 2.999714e-04 2.128694e-04 1.709659e-03 1.623825e-03 4.160121e-05 1.190050e-04 
##          259          260          261          262          263          264 
## 2.783962e-03 4.220669e-04 5.012555e-03 3.502698e-03 2.745375e-04 8.206397e-04 
##          265          266          267          268          269          270 
## 1.894728e-03 7.045750e-04 1.214010e-04 2.322880e-04 4.884182e-04 9.829011e-04 
##          271          272          273          274          275          276 
## 2.017807e-05 3.784961e-05 2.941640e-04 6.498525e-04 6.209947e-04 6.882779e-05 
##          277          278          279          280          281          282 
## 6.751263e-04 1.145093e-03 2.973867e-04 3.041667e-04 1.744043e-03 6.187570e-06 
##          283          284          285          286          287          288 
## 1.780876e-04 6.173502e-04 2.932442e-04 8.600843e-05 9.345685e-04 4.963507e-04 
##          289          290          291          292          293          294 
## 2.567321e-03 3.617525e-04 2.500105e-04 1.046003e-03 7.693488e-04 1.307944e-03 
##          295          296          297          298          299          300 
## 5.377909e-04 9.337509e-05 5.232381e-04 1.127785e-03 1.296490e-03 1.336911e-03 
##          301          302          303          304          305          306 
## 4.945008e-05 3.089426e-03 3.597263e-04 1.665548e-03 3.179473e-03 1.500078e-03 
##          307          308          309          310          311          312 
## 9.377717e-05 1.630285e-04 2.538308e-04 1.232696e-04 1.541966e-03 2.781060e-06 
##          313          314          315          316          317          318 
## 6.429360e-04 4.141661e-05 1.100340e-04 7.756757e-03 6.266110e-03 3.146947e-03 
##          319          320          321          322          323          324 
## 2.986093e-03 5.038954e-04 1.678454e-04 3.968006e-05 2.429374e-05 4.332314e-04 
##          325          326          327          328          329          330 
## 2.518175e-04 2.402268e-03 1.258893e-03 1.127178e-03 6.687371e-03 3.745045e-03 
##          331          332          333          334          335          336 
## 7.010791e-04 1.054133e-02 1.847345e-05 3.226082e-04 6.133326e-04 4.134558e-04 
##          337          338          339          340          341          342 
## 1.558257e-04 2.823642e-03 3.310190e-04 5.547554e-04 1.958648e-05 5.767271e-03 
##          343          344          345          346          347          348 
## 8.728290e-03 1.423261e-06 2.484673e-04 7.953327e-06 9.617142e-05 4.544587e-04 
##          349          350          351          352          353          354 
## 1.015887e-04 3.501337e-03 1.135030e-02 5.433935e-04 4.292583e-03 9.389302e-04 
##          355          356          357          358          359          360 
## 5.949221e-03 1.061641e-04 1.450082e-03 1.220245e-03 3.878350e-04 3.145547e-03 
##          361          362          363          364          365          366 
## 3.270204e-03 1.508866e-02 1.031074e-02 1.574193e-04 6.435982e-04 1.247757e-04 
##          367          368          369          370          371          372 
## 5.660949e-03 8.792416e-03 1.338518e-03 2.009602e-04 1.230509e-04 1.395403e-03 
##          373          374          375          376          377          378 
## 7.776009e-05 1.352619e-02 1.588136e-02 1.416125e-05 9.505888e-03 1.764746e-04 
##          379          380          381          382          383          384 
## 2.026535e-02 2.866143e-04 8.227437e-05 6.651736e-05 2.862637e-04 3.188983e-04 
##          385          386          387          388          389          390 
## 6.201179e-03 6.071435e-05 5.265106e-05 9.453468e-05 5.591786e-04 2.708232e-02 
##          391          392          393          394          395          396 
## 4.357551e-02 3.125491e-05 1.692657e-04 3.212243e-03 7.497023e-04 3.152622e-05 
##          397          398          399          400          401          402 
## 1.590388e-03 7.799167e-04 4.709193e-05 1.443526e-03 1.788712e-07 2.274145e-03 
##          403          404          405          406          407          408 
## 5.804156e-04 5.817042e-04 1.171026e-03 8.107716e-05 7.909289e-04 7.574811e-04 
##          409          410          411          412          413          414 
## 9.446013e-06 1.688684e-04 2.114634e-05 2.665259e-03 1.207304e-05 3.646793e-03 
##          415          416          417          418          419          420 
## 9.237106e-04 3.162309e-03 5.602221e-04 2.031118e-05 1.719697e-03 2.524083e-03 
##          421          422          423          424          425          426 
## 3.008282e-03 1.253466e-04 1.444209e-03 2.602724e-04 1.400520e-04 4.347142e-04 
##          427          428          429          430          431          432 
## 1.107302e-03 1.148027e-02 1.319521e-02 3.680071e-06 5.061113e-04 4.493698e-04 
##          433          434          435          436          437          438 
## 1.895808e-03 1.107297e-03 9.803797e-03 1.497241e-04 4.605079e-04 2.152513e-04 
##          439          440          441          442          443          444 
## 2.944386e-05 1.612511e-03 3.362744e-04 2.884537e-03 4.733477e-05 1.090125e-05 
##          445          446          447          448          449          450 
## 9.589350e-05 5.936347e-03 4.982653e-05 9.679593e-05 2.857653e-04 3.566635e-03 
##          451          452          453          454          455          456 
## 3.253391e-02 1.121974e-03 5.118650e-05 4.217875e-03 2.827604e-04 1.339141e-04 
##          457          458          459          460          461          462 
## 1.046473e-03 2.620268e-08 8.125263e-03 1.991486e-03 7.178070e-05 5.918874e-04 
##          463          464          465          466          467          468 
## 1.273682e-04 4.135719e-03 2.853847e-04 3.173735e-04 2.458840e-03 1.313956e-03 
##          469          470          471          472          473          474 
## 5.780277e-05 1.071080e-03 4.774297e-06 6.695363e-04 4.092412e-04 1.927520e-04 
##          475          476          477          478          479          480 
## 3.077767e-04 2.124336e-05 5.195115e-03 8.677694e-03 1.980321e-04 3.224312e-04 
##          481          482          483          484          485          486 
## 5.963074e-03 2.610010e-03 1.600034e-03 1.997672e-03 2.804816e-04 3.014607e-06 
##          487          488          489          490          491          492 
## 4.418145e-05 3.321042e-04 1.240100e-03 1.013186e-03 1.473349e-04 6.932471e-04 
##          493          494          495          496          497          498 
## 5.236953e-04 7.933896e-04 5.775743e-04 8.511641e-04 4.371039e-05 5.185947e-04 
##          499          500 
## 7.585227e-04 9.712721e-04
hatvalues(mfit) 
##           1           2           3           4           5           6 
## 0.013549287 0.005533546 0.008275536 0.003277853 0.008287643 0.004841357 
##           7           8           9          10          11          12 
## 0.003580605 0.006764909 0.006690287 0.004820021 0.006746627 0.013207949 
##          13          14          15          16          17          18 
## 0.009341717 0.005459476 0.003944172 0.022001529 0.008614707 0.020825451 
##          19          20          21          22          23          24 
## 0.024624657 0.007853395 0.004189364 0.004011857 0.013943151 0.005279961 
##          25          26          27          28          29          30 
## 0.007278029 0.010518195 0.016522072 0.006085409 0.011215737 0.004154521 
##          31          32          33          34          35          36 
## 0.005575054 0.006055264 0.010168984 0.009292679 0.003072573 0.002304438 
##          37          38          39          40          41          42 
## 0.042918505 0.012423115 0.007861509 0.015122035 0.009675679 0.015868614 
##          43          44          45          46          47          48 
## 0.005541902 0.015648840 0.004449012 0.004312333 0.009728066 0.007404931 
##          49          50          51          52          53          54 
## 0.027369611 0.005507429 0.010207787 0.008305879 0.008676827 0.011736780 
##          55          56          57          58          59          60 
## 0.004177757 0.006378689 0.005442759 0.013360041 0.009299951 0.003646747 
##          61          62          63          64          65          66 
## 0.018263033 0.007807814 0.006386336 0.005983582 0.004193610 0.014889241 
##          67          68          69          70          71          72 
## 0.030635730 0.010726449 0.007761921 0.004285364 0.007396787 0.016814831 
##          73          74          75          76          77          78 
## 0.008558472 0.007169741 0.004075364 0.005967758 0.009915403 0.007116486 
##          79          80          81          82          83          84 
## 0.012955256 0.006165736 0.004763520 0.006517307 0.017983905 0.004147247 
##          85          86          87          88          89          90 
## 0.006221637 0.005615599 0.017720440 0.004432751 0.017797040 0.003781632 
##          91          92          93          94          95          96 
## 0.006189212 0.005277586 0.012047497 0.011684771 0.019231965 0.002931971 
##          97          98          99         100         101         102 
## 0.006070866 0.005771692 0.007366416 0.015766572 0.004642524 0.005605395 
##         103         104         105         106         107         108 
## 0.009529855 0.003399468 0.009906385 0.012655299 0.014251882 0.014545166 
##         109         110         111         112         113         114 
## 0.009149451 0.007866390 0.003313011 0.014676418 0.005166948 0.007012010 
##         115         116         117         118         119         120 
## 0.006408809 0.006688242 0.024042080 0.014396780 0.005469329 0.004447327 
##         121         122         123         124         125         126 
## 0.008306883 0.010147812 0.008722320 0.012275340 0.003962253 0.008433943 
##         127         128         129         130         131         132 
## 0.005774947 0.012383190 0.018724941 0.008999210 0.009477288 0.006344975 
##         133         134         135         136         137         138 
## 0.006303371 0.015034033 0.005390477 0.005443776 0.022620035 0.006803268 
##         139         140         141         142         143         144 
## 0.011827431 0.016903852 0.003657262 0.004161823 0.007617547 0.009966052 
##         145         146         147         148         149         150 
## 0.009555097 0.007040490 0.005557434 0.005184403 0.007029120 0.008548986 
##         151         152         153         154         155         156 
## 0.003773020 0.038859119 0.015376674 0.008058520 0.006985657 0.009753832 
##         157         158         159         160         161         162 
## 0.020273086 0.013518197 0.005719993 0.003995290 0.008114050 0.011914744 
##         163         164         165         166         167         168 
## 0.003504814 0.031704771 0.009186174 0.008746894 0.007604521 0.008864814 
##         169         170         171         172         173         174 
## 0.020686459 0.007986693 0.006640899 0.008723107 0.012241344 0.004792426 
##         175         176         177         178         179         180 
## 0.016889379 0.007138168 0.005780097 0.009013190 0.005723565 0.020854744 
##         181         182         183         184         185         186 
## 0.010880441 0.006992715 0.004050242 0.009027107 0.010189438 0.059033495 
##         187         188         189         190         191         192 
## 0.020444284 0.009865370 0.007556495 0.003114736 0.004905710 0.003005777 
##         193         194         195         196         197         198 
## 0.008608843 0.007019496 0.006227746 0.007717805 0.004310056 0.007502887 
##         199         200         201         202         203         204 
## 0.007704636 0.004654252 0.009232956 0.007337941 0.016970456 0.004335310 
##         205         206         207         208         209         210 
## 0.009086970 0.007707000 0.004613985 0.022416442 0.012189916 0.010247388 
##         211         212         213         214         215         216 
## 0.004449329 0.009681078 0.016251369 0.008265622 0.007973695 0.004719761 
##         217         218         219         220         221         222 
## 0.017666285 0.004530755 0.003710460 0.006735481 0.009132436 0.007915835 
##         223         224         225         226         227         228 
## 0.008158760 0.005544232 0.004848946 0.042956432 0.004038641 0.013780066 
##         229         230         231         232         233         234 
## 0.006484066 0.014256791 0.018930697 0.009487834 0.010238706 0.015884138 
##         235         236         237         238         239         240 
## 0.005563522 0.006179603 0.011172211 0.009276987 0.008659431 0.005346606 
##         241         242         243         244         245         246 
## 0.015954141 0.004241114 0.017106752 0.005314971 0.002725421 0.006657994 
##         247         248         249         250         251         252 
## 0.007271747 0.012927397 0.009681741 0.011962197 0.008714474 0.010417292 
##         253         254         255         256         257         258 
## 0.005124933 0.003878998 0.008794482 0.008904300 0.005698989 0.009920229 
##         259         260         261         262         263         264 
## 0.005559650 0.004399023 0.006612674 0.009986594 0.007055918 0.006359001 
##         265         266         267         268         269         270 
## 0.004036095 0.003950984 0.006385511 0.008670387 0.007617414 0.003874705 
##         271         272         273         274         275         276 
## 0.005213672 0.005011292 0.004909178 0.003465050 0.007103864 0.005818190 
##         277         278         279         280         281         282 
## 0.004284623 0.008948348 0.009996440 0.004701196 0.006634092 0.016774505 
##         283         284         285         286         287         288 
## 0.006178054 0.003936398 0.010867483 0.005744833 0.007542923 0.005412542 
##         289         290         291         292         293         294 
## 0.007965899 0.014169840 0.008180637 0.005703621 0.004583747 0.006877021 
##         295         296         297         298         299         300 
## 0.019869757 0.004531595 0.019115197 0.008665590 0.006834021 0.012256809 
##         301         302         303         304         305         306 
## 0.009124936 0.007949045 0.004249845 0.010404923 0.009216320 0.006677807 
##         307         308         309         310         311         312 
## 0.004405537 0.005923429 0.008536047 0.007009244 0.007168952 0.003940313 
##         313         314         315         316         317         318 
## 0.004345075 0.005277856 0.005314482 0.017434886 0.021079591 0.007680251 
##         319         320         321         322         323         324 
## 0.010806961 0.013944878 0.007649566 0.023827642 0.007258786 0.010346457 
##         325         326         327         328         329         330 
## 0.009846838 0.036979370 0.011250579 0.006049176 0.010326927 0.014199390 
##         331         332         333         334         335         336 
## 0.004803179 0.030586120 0.007402210 0.007906887 0.008860450 0.005649074 
##         337         338         339         340         341         342 
## 0.004122687 0.011916705 0.007793461 0.005682006 0.006145419 0.009262543 
##         343         344         345         346         347         348 
## 0.027858954 0.004911526 0.007084146 0.004553217 0.007207065 0.005187095 
##         349         350         351         352         353         354 
## 0.007928047 0.021903171 0.035048239 0.008718513 0.027008109 0.007543925 
##         355         356         357         358         359         360 
## 0.010517870 0.003131693 0.032032583 0.004842367 0.010579480 0.020561609 
##         361         362         363         364         365         366 
## 0.007085815 0.023944033 0.012862678 0.006069949 0.005948138 0.005253157 
##         367         368         369         370         371         372 
## 0.013206871 0.004638247 0.015918347 0.010814470 0.004159656 0.007065616 
##         373         374         375         376         377         378 
## 0.005382047 0.037759459 0.009235761 0.003252706 0.014015037 0.004636161 
##         379         380         381         382         383         384 
## 0.021496628 0.003022836 0.006127400 0.005923398 0.010409399 0.004551968 
##         385         386         387         388         389         390 
## 0.019167899 0.018961970 0.007829671 0.007695704 0.024392840 0.021932645 
##         391         392         393         394         395         396 
## 0.024129134 0.003717459 0.004922214 0.018689358 0.009805939 0.006888865 
##         397         398         399         400         401         402 
## 0.009872677 0.006561849 0.005611843 0.005707463 0.007251774 0.006272782 
##         403         404         405         406         407         408 
## 0.008832765 0.011010763 0.010666405 0.004921823 0.011037377 0.003839243 
##         409         410         411         412         413         414 
## 0.015945135 0.009615193 0.002579398 0.013711094 0.008291139 0.009941848 
##         415         416         417         418         419         420 
## 0.011565889 0.015703399 0.031145502 0.003509472 0.008008397 0.012458793 
##         421         422         423         424         425         426 
## 0.008560414 0.003886298 0.007295455 0.015527392 0.011379947 0.004562744 
##         427         428         429         430         431         432 
## 0.012144682 0.007804892 0.027327041 0.012024145 0.005549796 0.005382785 
##         433         434         435         436         437         438 
## 0.006746420 0.014223385 0.012667765 0.005118210 0.023394486 0.016279756 
##         439         440         441         442         443         444 
## 0.008548915 0.004945220 0.003758112 0.015012651 0.014629007 0.011895734 
##         445         446         447         448         449         450 
## 0.009004568 0.011431527 0.016419347 0.008033024 0.008517971 0.012301981 
##         451         452         453         454         455         456 
## 0.018096806 0.012703645 0.002525320 0.008293851 0.004875099 0.005429912 
##         457         458         459         460         461         462 
## 0.003780246 0.016445476 0.020382582 0.022715924 0.004364339 0.003979695 
##         463         464         465         466         467         468 
## 0.004222459 0.005291461 0.005749860 0.017427273 0.013494739 0.007424092 
##         469         470         471         472         473         474 
## 0.013008650 0.007163236 0.006570245 0.003638937 0.009304590 0.014742078 
##         475         476         477         478         479         480 
## 0.003519339 0.002368021 0.008507498 0.020630183 0.007751670 0.007790989 
##         481         482         483         484         485         486 
## 0.028734611 0.006178132 0.024743541 0.011385560 0.012428093 0.005315830 
##         487         488         489         490         491         492 
## 0.004247147 0.016466570 0.009005512 0.004502619 0.006243791 0.013343254 
##         493         494         495         496         497         498 
## 0.005856979 0.012074787 0.010920007 0.017134803 0.003181510 0.024643230 
##         499         500 
## 0.028510555 0.009515486